| L(s) = 1 | + 2-s + 4-s − 3·5-s + 7-s + 8-s − 3·10-s − 3·11-s + 14-s + 16-s − 17-s + 8·19-s − 3·20-s − 3·22-s + 3·23-s + 4·25-s + 28-s + 9·29-s − 4·31-s + 32-s − 34-s − 3·35-s + 2·37-s + 8·38-s − 3·40-s − 6·41-s + 5·43-s − 3·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.34·5-s + 0.377·7-s + 0.353·8-s − 0.948·10-s − 0.904·11-s + 0.267·14-s + 1/4·16-s − 0.242·17-s + 1.83·19-s − 0.670·20-s − 0.639·22-s + 0.625·23-s + 4/5·25-s + 0.188·28-s + 1.67·29-s − 0.718·31-s + 0.176·32-s − 0.171·34-s − 0.507·35-s + 0.328·37-s + 1.29·38-s − 0.474·40-s − 0.937·41-s + 0.762·43-s − 0.452·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 361998 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 361998 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.503779824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.503779824\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | Isogeny Class over $\mathbf{F}_p$ |
|---|
| bad | 2 | \( 1 - T \) | |
| 3 | \( 1 \) | |
| 7 | \( 1 - T \) | |
| 13 | \( 1 \) | |
| 17 | \( 1 + T \) | |
| good | 5 | \( 1 + 3 T + p T^{2} \) | 1.5.d |
| 11 | \( 1 + 3 T + p T^{2} \) | 1.11.d |
| 19 | \( 1 - 8 T + p T^{2} \) | 1.19.ai |
| 23 | \( 1 - 3 T + p T^{2} \) | 1.23.ad |
| 29 | \( 1 - 9 T + p T^{2} \) | 1.29.aj |
| 31 | \( 1 + 4 T + p T^{2} \) | 1.31.e |
| 37 | \( 1 - 2 T + p T^{2} \) | 1.37.ac |
| 41 | \( 1 + 6 T + p T^{2} \) | 1.41.g |
| 43 | \( 1 - 5 T + p T^{2} \) | 1.43.af |
| 47 | \( 1 + 6 T + p T^{2} \) | 1.47.g |
| 53 | \( 1 - 6 T + p T^{2} \) | 1.53.ag |
| 59 | \( 1 + 6 T + p T^{2} \) | 1.59.g |
| 61 | \( 1 + 4 T + p T^{2} \) | 1.61.e |
| 67 | \( 1 - 5 T + p T^{2} \) | 1.67.af |
| 71 | \( 1 + p T^{2} \) | 1.71.a |
| 73 | \( 1 - 8 T + p T^{2} \) | 1.73.ai |
| 79 | \( 1 + 10 T + p T^{2} \) | 1.79.k |
| 83 | \( 1 - 12 T + p T^{2} \) | 1.83.am |
| 89 | \( 1 + 6 T + p T^{2} \) | 1.89.g |
| 97 | \( 1 - 8 T + p T^{2} \) | 1.97.ai |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.40035716279358, −12.04655307038836, −11.71068095761863, −11.23265382674556, −10.96914851553707, −10.30621149979847, −10.02987326290166, −9.244307343117516, −8.840309173284464, −8.113104914794551, −7.942359109256884, −7.461019964985191, −7.057213157734263, −6.583191648160984, −5.885441762568334, −5.318690383010742, −4.939451973191510, −4.586220438009238, −3.992465035407060, −3.371171782627621, −3.095198890322810, −2.549560036787147, −1.788216791916376, −1.007654523908982, −0.4917864216136490,
0.4917864216136490, 1.007654523908982, 1.788216791916376, 2.549560036787147, 3.095198890322810, 3.371171782627621, 3.992465035407060, 4.586220438009238, 4.939451973191510, 5.318690383010742, 5.885441762568334, 6.583191648160984, 7.057213157734263, 7.461019964985191, 7.942359109256884, 8.113104914794551, 8.840309173284464, 9.244307343117516, 10.02987326290166, 10.30621149979847, 10.96914851553707, 11.23265382674556, 11.71068095761863, 12.04655307038836, 12.40035716279358
